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Periodic and homoclinic solutions to a class of Hamiltonian systems with indefinite potential in sign. (English) Zbl 1013.34038
The author studies nonautonomous second-order systems of the form $$\ddot x- A(t) x+b(t) V'(x)= 0,$$ where $b$ is a continuous, $T$-periodic real function which may change sign, $A$ is a continuous, $T$-periodic positive definite matrix-valued function and $V\in C^2(\bbfR^n, \bbfR)$ satisfies a superquadratic growth condition. Under different sets of additional technical conditions, the author proves the existence of a nontrivial $T$-periodic solution, the existence, for any natural number $k$, of a nontrivial $kT$-periodic solution and the existence of one homoclinic solution. The proofs rely on variational arguments. More specifically, the periodic solutions are found as critical points of the action functionals $$f_k(u)= \int^{kT}_0 \Biggl[{1\over 2}|\dot u|^2+ {1\over 2}\langle A(t) u,u\rangle- b(t) V(u)\Biggr] dt$$ on the spaces $\{u\in H^1((0,kT), \bbfR^n): u(0)= u(kT)\}$, through an application of the mountain pass theorem; the homoclinic solution is found as the limit for $k\to\infty$ of the $kT$-periodic solutions. Lagrangian systems with a potential changing sign have been considered, e.g., in papers by {\it L. Lassoued} [Ann. Mat. Pura Appl., IV. Ser. 156, 73-111 (1990; Zbl 0724.34051)] and {\it M. Girardi} and {\it M. Matzeu} [NoDEA, Nonlinear Differ. Equa. Appl. 2, No. 1, 35-61 (1995; Zbl 0821.34041)] in connection with the search for periodic solutions, and by {\it P. Caldiroli} and {\it P. Montecchiari} [Commun. Appl. Nonlinear Anal. 1, No. 2, 97-129 (1994; Zbl 0867.70012)] in connection with the search for homoclinic solutions.
Reviewer: Maria Letizia Bertotti (MR 97e:58040)

34C25Periodic solutions of ODE
37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods